isAlgClosure_of_perfectField — Mathlib

English

Given K, L, E with tower K ≤ L ≤ E and L/E separable, every K-algebra hom E → M arises as a restriction from some E → M that factors through L.

Русский

При наличии каскада полей K ⊆ L ⊆ E и L/E сепарабельности, любой K-алгебромом E→M получаемый как RestrictDomain L является ограничением некоторого гомоморфа через L.

LaTeX

$$Surjective (AlgHom.restrictDomain L)$$

Lean4

/-- If `k` is perfect, `K` is a separable closure of `k`,
then it is also an algebraic closure of `k`. -/
instance (priority := 100) isAlgClosure_of_perfectField [Algebra k K] [IsSepClosure k K] [PerfectField k] :
    IsAlgClosure k K :=
  have halg : Algebra.IsAlgebraic k K := IsSepClosure.separable.isAlgebraic
  haveI := halg.perfectField;
  inferInstance

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